is there a classification/characterization of fields K for which the automorphism group Aut(K) has the property that |Aut(K)| < |K| (e.g. finite fields, the rationals and reals) ?
What about the same question for real-closed fields K ?
Many thanks ...

@Gerhard: Unfortunately, the reals have lots of vector-space automorphisms over the prime field $\mathbb Q$ but no nontrivial field automorphisms. (I like to see things turn into set theory, but this one will need more work.)
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Andreas BlassMay 10 '12 at 16:03

2

The fields ${\mathbb{Q}}_p$ have only the identity automorphism, like the reals.
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LubinMay 10 '12 at 16:31

Gerhard, I guess we are using the terminology "prime field" with different meanings. To me, it means a field with no proper subfields, i.e., either the rationals or a finite field of prime order.
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Andreas BlassMay 11 '12 at 18:46